No, the Doctrines need NOT be equipped on the specific Weapon to be effective. You can have a doctrine on any item in order of it to be effective/in effect. But you need to have it in the open in order for it to have effect. With that I mean, you need to make sure it is being used in according to which Weapon Mode you are currently in.

With this I mean, (but I´m not 100% sure) if you are having a Doctrine on Weapon in Mode 1, and if you are using Weapon in Mode 2, the Doctrine(s) does not count towards the one you are currently using. This is at least the case for Signums.

"However I'm playing on PS4..."

If you're playing on console, some of my response doesn't apply. I don't believe the keyword "supreme" has been added yet, so figuring out which effects exceed the soft cap is going to be more difficult for you.

"...I can't hover over the stats to see tooltip information (not that I know of a way to do that anyway)."

PC also supports a controller, so I checked it out. Pressing 'select' (maybe "back" for you or whatever the button next to "start" is) and then using the left stick or D-pad to navigate around lets you view the tooltips for the character sheet. My guess is that consoles work the same way.

"Psalm Codes still apply even when you hit the soft or hard cap? Or only when you hit te soft cap?"

No, the caps apply evenly across the board. Psalm codes are not really that different than gear enchants. For example, some relic enchants stack (e.g, "X% Damage Bonus in Enraged state") and some don't (e.g., "When in Enraged state enemies within 5 meters are Feared"). Some psalm codes stack (e.g., "+10% DMG on Bleeding enemies") and some don't ("When struck, gain +20% damage for 2 seconds."). But all enchants and psalm codes obey the caps. Some enchants and psalm codes are limited by the hard cap (in the PC version, these are all listed as giving 'supreme' stats, but in the console it's ambiguous until they add the keyword for you folks) but more are limited by the soft cap.

"I'll have a look at the compendium to see if I can figure out which psalm codes and doctrines stack."

Be prepared for some ambiguity. Some are explicitly listed, but it can't hurt to play around and test things. For instance, the Uncreator psalm sounds (from its description) like it wouldn't stack---after all, what does changing 10 to 15 multiple times do? But in reality, its behavior is actually adding 50% to vulnerability effectiveness, and it does stack. Inquisitor is not a game known for its good descriptions, unfortunately.

• Having the same psalm code socketed multiple times on  a piece of gear
  • Does this effectivly apply the bonus from that psalm code multiple times or is the bonus for that specific psalm code only taken into account once?

No. Only one psalm code per item will give a benefit.

• Having the same psalm code on different pieces of gear
  • I'm pretty sure this is correct but when you have for example Iron Psalm on your main implant and again on your armor: does it apply twice? It is my understanding that having the same psalm code spread over different pieces of gear means that psalm code's bonus gets applied multiple times

Yes, so long as the psalm code itself stacks (some do not, and usually the compendium lists that in the description).

• Can you have the same Psalm Doctrine active multiple times.
  • I have for example the following psalm doctrine active on my armor:
    50% extra physical damage and physical resistance for enemies is halved
    Can I apply the same doctrine on an other piece of gear and will that doctrine then be applied so basically I'll have two increases for physical damage and the resistance for enemies get halved twice?

Yes, so long as the doctrine itself stacks. Usually doctrines which give a +X bonus (damage, for instance) do stack. Usually doctrines which qualitatively change the behavior of something do not stack.

• Stat cap from bonusses
  • Not specifically related tot Psalm codes but related nonetheless.
    Let's say I have my physical damage bonus at 75%. If I then apply a 50% physical damage bonus from a Psalm code or doctrine, will it be capped at 100%? or will it increase to 125%
    I'd like to know what the caps are for the different stats (if such information is publicly known and shared) and if applying bonusses from equipment, psalm codes etc.. are limited to those caps or can go beyond those caps.

Psalm codes have no special behavior with respect to caps. If the stat itself has a cap (soft or hard), it will still apply. Soft caps (60% for resist, DR, deflect, and dodge; 50% for crit) are listed in the character sheet. Hover over the stat for more information. Soft caps cap be exceeded with "supreme" stat bonuses. Hard caps (80% for resist, DR, deflect, and dodge; 100% for crit; 75% for suppression resistance) are also listed in the character sheet tooltips, but can never be exceeded. Normally, if a stat is capped (soft or hard), it will appear yellow in the character sheet (suppression resistance is currently bugged and will show up as exceeding 75%, but it will not give any benefit over 75%).